let ξ(x) =Σ_(n=1) ^∞ (1/n^x ) with x>1 1) calculate lim_(x→1^+ ) ξ(x) and lim_(x→+∞) ξ(x) 2) prove that ξ(x) =1+2^(−x) +o(2^(−x) ) (x→+∞) 3) prove that ξ is decreasing and convexe fucntion on]1,+∞[


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Question Number 62425 by mathmax by abdo last updated on 21/Jun/19

$l e t ξ (x) = \sum_{n = 1}^{\infty} \frac{1}{n^{x}} w i t h x > 1$ $1) c a l c u l a t e {l i m}_{x \to 1^{+}} ξ (x) a n d {l i m}_{x \to + \infty} ξ (x)$ $2) p r o v e t h a t ξ (x) = 1 + 2^{- x} + o (2^{- x}) (x \to + \infty)$ $3) p r o v e t h a t ξ i s d e c r e a s i n g a n d c o n v e x e f u c n t i o n o n] 1, + \infty [$
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